Problem: Factor the following expression: $-8$ $x^2+$ $41$ $x$ $-5$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-8)}{(-5)} &=& 40 \\ {a} + {b} &=& & & {41} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $40$ and add them together. The factors that add up to ${41}$ will be your ${a}$ and ${b}$ When ${a}$ is ${1}$ and ${b}$ is ${40}$ $ \begin{eqnarray} {ab} &=& ({1})({40}) &=& 40 \\ {a} + {b} &=& {1} + {40} &=& 41 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-8}x^2 +{1}x +{40}x {-5} $ Group the terms so that there is a common factor in each group: $ ({-8}x^2 +{1}x) + ({40}x {-5}) $ Factor out the common factors: $ x(-8x + 1) - 5(-8x + 1) $ Notice how $(-8x + 1)$ has become a common factor. Factor this out to find the answer. $(-8x + 1)(x - 5)$